This most excellent elucidation of the disposition of this word appears in The American Heritage® Dictionary of the English Language, Third Edition. This blurb manages to avoid the nasty dilemma "Is it a word?" as JerboaKolinowski pointed out: "mistakenly believe to be correct ... in formal style" is both accurate and philosophically sound.
The label Non-Standard does only approximate justice to the status of irregardless. More precisely, it is a form that many people mistakenly believe to be a correct usage in formal style but that in fact has no legitimate antecedents in either standard or nonstandard varieties. (The word was likely coined from a blend of irrespective and regardless.) Perhaps this is why critics have sometimes insisted that there is “no such word” as irregardless, a charge they would not think of leveling at a bona fide nonstandard word such as ain't, which has an ancient genealogy.
This word is also a good example of why one cannot always trust on-line information sources: WordNet defines irregardless without so much as a word indicating the scorn one might incur should one use it in educated circles. (Enough people have messaged me their hate for WordNet, by the way. I'm sure there is somewhere more constuctive than my inbox to express that opinion.)